The development of Renewal Theory began in the 1940’s in operations research for the industry, and yet it found unprecedented applications in stochastic processes in many different areas of science since then. In this work we investigate and model stochastic processes that lead to renewal properties with no memory, both for Poisson statistics where the constituents act independently and non-Poisson statistics where the behavior stems from cooperative effects. We develop methods to determine the correlations between fluctuations and perturbing signal. The evolution of the degree of complexity for systems under linear and non-linear perturbations are analized and the process through which a memory emerges is revealed in otherwise memoryless systems. A review of Diffusion Entropy method for detecting the degree of complexity of statistical data is made and DEA methods are applied to the analysis of data for different statistics to see the interplay of complexity and perturbation. Renewal theory finds many applications in atmosphere physics, eartquakes, solar flares or neuronal data analysis, econophysics and recently the number of scientific applications are growing.